A comparison of the influence of plasmoid-drift mechanisms on plasma fuelling by cryogenic pellets in ITER and Wendelstein 7-X

Pellet injection is the most promising technique to achieve efficient plasma core fuelling, key for attaining stationary scenarios in large magnetic confinement fusion devices. In this paper, the injection of pellets with different volumes and speeds into standard plasma scenarios in ITER (tokamak) and Wendelstein 7-X (stellarator) is studied by modeling the pellet ablation and particle deposition, focusing on the evaluation of the expected differences in pellet plasmoid drifts in tokamaks and stellarators. Since the efficiency of the damping-drift mechanisms is predicted to depend on the magnetic configuration, device-specific characteristics are expected for the temporal evolution of the plasmoid drift acceleration. For instance, plasmoid-internal Pfirsch–Schlüter currents dominate the drift damping process for stellarators, while plasmoid-external currents are more relevant for tokamaks. Also, relatively larger drifts are in principle expected for W7-X due to higher field gradients in relation to machine dimensions. However, shorter plasmoid-internal charge reconnection lengths result in the drift damping due to internal Pfirsch–Schlüter currents being more effective than in a tokamak. Therefore, the average relative drift displacement during the whole plasmoid homogenization may a priori be comparable in both magnetic configurations. Moreover, High Field Side (HFS) injection is expected to be highly advantageous to maximize pellet particle deposition in ITER, whereas it may only be beneficial in medium to high β environments in W7-X. Finally, there may be means for the optimization of pellet injection configurations in both ITER and W7-X for the considered plasma scenarios despite the sizeable differences in the relative importance of the mechanisms of plasmoid drift acceleration and deceleration in play.


Introduction
An efficient fuelling capability is mandatory for stationary operation in large magnetic-confinement fusion devices, since high density is necessary to achieve the conditions required for fusion [1]. In particular, according to neoclassical theory, fuelling is especially critical for helical type devices due to core particle depletion for central heating. Cryogenic pellet injection is currently the best candidate for efficient fuelling, since penetrations achieved by pellets are much deeper than those for gas puffing. Indeed, pellet injection allows particle deposition inside the edge transport barrier, while gas puffing deposits the material at the plasma edge [2]. Both experiments and modeling show that fuelling efficiency, defined as the ratio between the number of deposited particles in the confined plasma and the number of injected particles, drops with shallowing of particle deposition [3][4][5]. Thus, the fuelling efficiency of pellet injection is much higher than that of gas puffing, especially for reactor scale plasmas with the Scrape of Layer opaque for penetration of neutrals from the edge [6]. In addition, the injection of a pellet does not have an associated energy source, which is the case for Neutral Beam Injection [2]. This additional energy source may be detrimental to density control, particularly for stellarators, due to energy and particle transport coupling that may lead to hollow density profiles, thereby worsening the situation [2]. In summary, although other techniques, such as supersonic gas injection or compact tori injection, should be contemplated, pellet injection can be considered as the prime candidate for the core fuelling system in next-step fusion devices such as the Demonstration Fusion Power Reactor (DEMO) [3]. Then, focusing on fuelling by pellets, the efficiency of their injection strongly depends on the ∇B-induced drift of the pellet material once it has been ionized. The magnetic configuration, among other quantities later described, determines such drift. Considering therefore the inherent differences of tokamaks and stellarators in terms of their magnetic fields, different deposition characteristics and fuelling efficiencies may be expected for these two families of devices. An analysis made using dedicated predictive pellet ablation and deposition modeling tools may help to reveal differences in drift mechanisms with the aim of determining the advantages of each configuration and use them for more effective pellet fuelling.
The processes of pellet ablation and subsequent material homogenization have been studied extensively by experiment during the past decades [7,8]. At present, experimental results are well explained by the different models describing the processes of pellet ablation [9][10][11][12][13][14] and plasmoid homogenization.
In particular, several homogenization models describe the expansion and the drift of the partially ionized cloud that detaches from a pellet as it is being ablated, i.e. plasmoid, in a self-consistent way, including different mechanisms that either enhance [15] or reduce [16][17][18] its drift acceleration. Since the main drift term, as well as the additional mechanisms, are magnetic-field dependent, the total plasmoid drift, and hence the final plasma density increase, depends on the magnetic field geometry. Indeed, in tokamaks, significantly different pellet particle deposition profiles are obtained when pellets are injected from the magnetic High-Field Side (HFS) rather than from the technically preferable Low-Field Side (LFS). In particular, for the former, deep deposition of the material and high fuelling efficiency can be achieved even when a pellet is completely ablated in the edge region [19][20][21][22]. Similarly, material drift down the magnetic field gradient is also expected for stellarators. Outward drifts are indeed observed for LFS injection [23,24]. However, experimental results obtained in the Large Helical Device, where HFS injections were carried out for the first time in a helical device, show that material drift is different from that observed for tokamaks [25,26]. More recently, first experiments in Wendelstein 7-X (W7-X), carried out with a blower-gun injector [27] during its OP 1.2 experimental campaign, appear to reaffirm this. Indeed, first results show no clear advantage for HFS injection [28,29], as opposed to observations in tokamaks, i.e. to date, inwards plasmoid drift is not clearly observed for HFS injections in W7-X. In contrast, simulations carried out for W7-X predict a measurable inwards drift for HFS injections [30]. The size and direction of the net drift displacement may depend on target plasma conditions and on the β dependence of the magnetic configuration under consideration. These dependencies need to be investigated in more detail in order to better comprehend the drift mechanisms at play. The consideration of the impact of the magnetic configuration is thus mandatory to optimize pellet fuelling efficiency.
In this work, drift mechanisms for two different types of magnetic field geometries, tokamak and stellarator, are analyzed and compared. The aim of this comparison is to evaluate the plasmoid drift when different damping effects dominate, which is the case for tokamaks and stellarators, due to the difference in the average plasmoid-internal and external selfreconnection lengths. For this, the HPI2 code [12,18,31] is used to simulate pellet injections into relevant plasma scenarios for ITER and W7-X. The HPI2 code can simulate both the ablation of a pellet and the homogenization of the ablated material in a consistent way. Pellet injections in ITER are additionally studied with the code simplified mass ablation and relocation treatment (SMART) [32], also able to simulate pellet ablation and homogenization. These two devices are chosen because they are closest to reactor parameters in both the stellarator and tokamak lines and thus can provide guidance for optimization of fuelling by pellets in future reactors based on both concepts.
In the paper, first the physics of pellet ablation and homogenization are described in section 2, and afterwards, the differences found between tokamaks and stellarators are highlighted. In section 3, the ablation and homogenization models used in the pellet source modeling codes HPI2 and SMART, along with some numerical assumptions, are described. Section 4 presents the results of the simulations of the pellet injection for ITER and W7-X scenarios. At the end of this section, the influence of the device configuration in the plasmoid drift is discussed. Finally, conclusions are presented in section 5.

Background: ablation and homogenization
A pellet injected into a magnetically confined plasma is ablated by incident background plasma particles, while the already ablated material shields the pellet from further interactions with the ambient plasma [7,8]. Subsequent to ablation and ionization, various phenomena occur until, finally, plasma temperature and density radial profiles recover their preinjection values. During the homogenization phase, detached plasmoids expand parallel to the magnetic field lines, until full plasmoid homogenization. At the same time, the potential distribution in the plasmoid is modified and the ablated material drifts down the magnetic field gradient [15,16,18]. These phenomena are not independent of each other since the parallel expansion of the plasmoid (pressure homogenization) reduces the drift acceleration and eventually stops it. In addition, some mechanisms, which originate in the winding of the magnetic field lines, affect the drift acceleration, and hence, can halt it before pressure equilibrium does so. Needless to say, the magnitudes of such mechanisms depend on the particular magnetic configuration.
The drift of the ablated material determines the shape of the deposition profile and the efficiency of the fuelling. The basic mechanism underlying the drift is the motion of plasmoid electrons and ions in the background inhomogeneous magnetic field. This motion originates an uncompensated curvature current, j∇ B , which for an arbitrary magnetic configuration is [26]: Here p 0, p ∞ are the pressures of the plasmoid and the background plasma, respectively, and B ∞ is the background magnetic field. In the case of a tokamak-like magnetic field of major radius R, this reduces to [33,34] This curvature current is balance by an ion polarization current, j pol : Here, n 0 is the plasmoid density, m i is the ion mass and E is a charge-induced electric field. From the condition of current closure, dE dt is determined. Therefore, for arbitrary magnetic configurations the main term of the cloud drift acceleration is [26,31] while it has the following form for the case of equation (2): Thus, due to plasmoid polarization, plasmoid particles are accelerated down the magnetic field gradient. In this simplified picture, the plasmoid drift acceleration, coupled to plasmoid parallel expansion, cannot stop before pressure equilibrium, i.e. when p 0 − p ∞ , as seen in equations (4) and (5). However, in order to better explain experimental observations [15,20,35], the previously mentioned additional effects, which either enhance or reduce the drift acceleration, must be included. Hence, the drift velocity is determined by the balance of these driving and damping drift terms [8]. The former term results from the compensation of the curvature current j∇ B by the polarization current j pol ; it also includes a driving term that arises due to the plasmoid parallel expansion, which enhances the curvature drift, and hence, the cloud polarization [15]. The latter term is related to the emission of an Alfvén wave that propagates from both ends of the plasmoid in the parallel direction. In addition to the Alfvén emission, two damping mechanisms, which lead to the deceleration of the drift due to the reduction of the plasmoid polarization by either plasmoid-internal or external Pfirsch-Schlüter currents, are considered. Plasmoid-internal currents arise in order to balance the variation of current accumulation on the plasmoid surface due to the rotation of magnetic field lines around the plasmoid center [16,17]. This damping effect, in the following referred as the Internal Circuit Closure (ICC) effect, is important for plasmoids with very long parallel lengths or for short plasmoid-internal charge reconnection lengths. On the other hand, the external current arises when, due to the finite plasmoid cross-section, regions of opposite polarization start to become interconnected along the magnetic field lines [18]. Thus, the cloud polarization, and hence, the plasmoid drift, are reduced. This effect, which has been evidenced in [36] through its correlation with the distribution of rational magnetic flux surfaces, will be referred to as the External Circuit Closure (ECC) effect. It depends also on the poloidal position of the plasmoid [18,31], since the distribution of charges inside the plasmoid can be asymmetric with respect to the tangent to the flux surface and, for this case, the fraction of reconnected charges may decrease.
The total displacement of the plasmoid, i.e. the distance, along the magnetic gradient direction (twofold integration of equation (4)), covered by the plasmoid from the detachment position until the drift is stopped, depends on the combined effect of the previously mentioned mechanisms. Since the additional mechanisms arise due to the winding of the magnetic field lines, it is expected that the dominant mechanism depends on the characteristics of the magnetic field structure. Hence, differences in the drift displacements are expected for non-axisymmetric devices with respect to what is observed in tokamaks [15]. Indeed, the drift-damping produced by external currents is the dominant effect in tokamaks, whereas internal currents are dominant in helical configurations due to their shorter plasmoid-internal charge reconnection lengths [26]. Thus, it is important to understand and quantify the actual effect of having different dominant damping-drift mechanisms, in order to optimize the fuelling injection in future reactors since plasmoid drift is a key element for material deposition and fuelling efficiency. So, for modeling purposes, comparing simulations undertaken for both types of devices with the same code adapted for each set-up is necessary. For this, the HPI2 code, which has been developed for the prediction of the pellet particle source in tokamaks [12,18,31] and can also be applied to devices with non-axisymmetric magnetic field configuration [26,30], is used to compare the plasmoid drift in both type of machines for the first time, using real magnetic configurations and geometrical parameters. In addition, the SMART code, that has been developed and applied for the modeling of pellet particle sources in a range of tokamak configurations [32], is used for to study pellet behavior in ITER. Thus, the intention here is to extend current knowledge regarding the suitability of each device as a reactor from the point of view of optimized pellet fuelling.

Plasmoid drift mechanisms in HPI2 and SMART
The HPI2 code simulates the ablation of a pellet and the homogenization of its ablated material in a consistent way, while considering the specific geometrical data, the plasma density and temperature profiles as well as the magnetic configuration of a particular device (see figure 1) [12,18,31]. First, for each position considered along its injection trajectory, the ablation rate is calculated using an enhanced version of the Neutral Gas and Plasma Shielding model [12,31], taking into account the plasma particle distribution by separating it into discrete groups of mono-energetic beams and following each beam through all the layers surrounding the pellet. Plasmoid dimensions and internal parameters are also obtained; they are used later as input for the homogenization calculations. Afterwards, the evolution of the plasmoid, once it is detached from the pellet, i.e. the plasmoid evolves independently from the pellet, is followed using a two-cell four-fluid Lagrangian model [18,31,37], by considering mass and energy conservation laws. In addition, plasmoid drift is calculated taking into consideration the fact that the drift and the plasmoid expansion are coupled, since the drift depends on the difference between plasmoid and plasma pressures. Therefore, the motion of the plasmoid barycenter is calculated self-consistently with the evolution of the plasmoid and the plasma [18,31]. The complete picture of the plasmoid evolution is given when the additional damping mechanisms are included in the calculations (see appendix A for more details). The HPI2 homogenization model includes the main drift term, proportional to the plasmaplasmoid pressure difference (in equation (4)), the curvature enhancement provoked by plasmoid parallel expansion, the damping effect of Alfvén wave emission, and the ICC and ECC effects. In addition, the momentum transfer from the plasmoid to the background plasma is taken into account. This code has been validated for pellet injection cases in a number of tokamaks, such as JET [38][39][40][41][42] and Tore Supra [35,36,43], as well as some other ones [18,44,45]. Moreover, it has been used to simulate pellet injection in a reduced number of nonaxisymmetric devices [26,29,30,46]. Furthermore, HPI2 has also been employed for pellet core fuelling studies in EU-DEMO. For instance, in combination with the automated system for transport analysis (ASTRA) transport code [47], to optimize the Pellet Injection System (PIS) set-up [48].
For this work, it is important to mention that, due to the differences in the magnetic field configurations, some simulation assumptions are not identical for both types of devices. For instance, for stellarators, being non-axisymmetric, the fact that the magnetic field and its gradient can vary strongly along the magnetic field lines must be taken into account. Therefore, the general drift equation for such devices is based on equation (6) instead of (4), i.e. the quantities | ⃗ B ∞ ||, | ⃗ B ∞ | 2 and |∇| ⃗ B ∞ | at the plasmoid center in equation (4) are replaced by the average over its length: An additional difference in the numerical assumptions is related to the calculation of the plasmoid barycenter. In the case of tokamaks, the direction of the drift displacement is assumed to be the same in (R, Z) coordinates for every toroidal coordinate, i.e. for all plasmoid particles. On the contrary, for stellarators, it should be considered that, as has already been mentioned, the internal short-circuit currents can dominate the drift damping process. Therefore, the rotation of E ∞ inside the plasmoid due to the ICC effect must be considered. In addition, a further simplification in the calculation of the drift is required. In particular, an effective drift for the entire plasmoid is assumed, by taking a toroidal average of ∇B, although, due to the variation of ∇B along the field lines, the local drift of a plasmoid is a function of the toroidal angle. This means that the effective drift is considered to be the same for all plasmoid particles in (ρ, θ) coordinates, regardless their toroidal Simplified diagram outlining the workflow of the HPI2 code, which simulates pellet ablation and plasmoid evolution by considering specific geometrical data, plasma profiles and the magnetic field configuration of the device. Top-left (yellow), local ablation rates are calculated which lead to initial plasmoid parameters. It is considered that a plasmoid detaches from the pellet when the ionization degree, f i , is high enough, i.e. f i ⩾ 0.95. Bottom-right (green), plasmoid time evolution is obtained with a two-cell four-fluid (plasma and plasmoid electrons and ions) Lagrangian model for fixed background plasma profiles, which leads to the deposition position of a single plasmoid. Left side (blue), plasma density and temperature profiles are updated (the global process is assumed to be adiabatic, i.e. the local decrease in temperature is proportional to the local increase in density). Updated profiles are used, along with updated pellet size, to calculate the local ablation rate for the new pellet position, which is calculated assuming constant pellet speed 8 . position; here, ρ is the normalized minor radius, while θ is the poloidal angle. The radial and poloidal components of such a drift are directly calculated by twofold integration of equation (6). Finally, the toroidal variation of ∇B causes a widening of the deposition profile, which is also taken into account. 8 There are several time scales at play in the process of pellet ablation and deposition: • Time interval between the deposition of two successive plasmoids, t dep , which is about 10 µs. • The homogenization and drift time for a single plasmoid, t hom , which is takes about 100 µs. • The total ablation and homogenization time of a pellet, t abl , which typically is approximately 1 ms. • The time for a global plasma response due to the radial transport, ttrs, with a minimum of several milliseconds.
It should be noted that it is assumed that a plasmoid is fully homogenized in the plasma when calculating the ablation rate and the plasmoid evolution at the next step, although in reality t dep is smaller than t abl . In addition, the total amount of material deposited in the plasma is equal in all cases to the pellet particle content multiplied by the calculated fueling efficiency (proportion of particles expelled out of the discharge during the drift).
In the SMART code [32], scaling-based models are in use for the description of the ablation and plasmoid drift processes allowing for a fast and efficient computation of the pellet particle source profile. Details of processes happening at time scales shorter than 0.1-1 ms are not considered in this model. Therefore, it does not deal with the specifics of pellet ablation or plasmoid formation and detachment. The ablation rate is described by a parameterization derived from a model by Kuteev [49], in which the energy distribution of incident particles as well as electrostatic sheath effects are taken into account. The plasmoid dimensions are calculated on basis of formulae obtained from a 1D steady-state flow model [50]. For the calculation of the drift displacement, the curvature and polarization currents (1) and (3) are assumed to be closed by the propagation of Alfvén currents in toroidal direction at short time scales. The drift displacement of a plasmoid is calculated using the simplified expression of [51] for the point of the force balance, dV/dt = 0. This displacement is assumed to be irreversible due to the possible dissipation processes described in [32]. In addition, both pellet ablation and particle displacement, or relocation, depend on the original pellet parameters and local plasma parameters at the pellet location. More details about these expressions can be found in appendix B. Validation versus experiments confirms rather accurate description of the outward drift for LFS injection as well as location of the deposition maxima, both for LFS and HFS injections, but could underestimate the inward widening of the particle deposition profile for the HFS injection.

Simulation results
In order to evaluate the suitability of tokamaks and stellarators for optimum pellet fuelling, it is necessary to understand and quantify the importance of the drift-damping mechanisms such as the ICC and ECC, in each type of devices. For that, one tokamak and one stellarator have been chosen to perform pellet simulations for relevant scenarios in both machines. Such a study can also be useful as an evaluation tool for fuelling performance in future fusion reactors, and complement others, as the already mentioned work from [48]. On the tokamak side, ITER is the chosen device, while W7-X is the selected stellarator [52]. ITER is chosen due to its relevance in the tokamak path towards a fusion power plant. It is designed to have energies and densities close to those of a reactor, and it is expected that it will demonstrate alpha heating dominated plasmas, i.e. Q ⩾ 10. Similar arguments are used for the chosen stellarator. W7-X is a large and optimized stellarator, with expected high performance, albeit a DD equivalent Q = 1 is not planned to be achieved. In this work, only single pellet injections have been considered in the simulations, although, in reality, series of pellets will be injected. Pellet series cannot be treated as easily as single-pellet injections since plasma evolution after one pellet injection must be correctly simulated before the injection of the subsequent pellet.

ITER
The ITER tokamak is designed to operate at the high densities, n e ∼ 10 20 m −3 required to achieve a target in fusion gain of Q ⩾ 10 [53]. These densities will be achieved by repetitive pellet fuelling because gas fuelling is expected to be very inefficient due to large plasma volume, ∼820 m 3 , and the high edge temperatures. The ITER PIS produces cylindrical pellets (from pure D to 10%D:90%T mixes) [54,55]. The ITER PIS design foresees four standard sizes of pellets, with pellet volumes, V p , of 92 mm 3 , 50 mm 3 , 33 mm 3 and 17 mm 3 , tuneable in the range ±20%, and maximal intact pellet speed of v p = 300 m s −1 (note that at the maximal design speed, v p = 500 m s −1 , there is a risk for the disintegration of the majority of pellets in the guide tube [56]). In addition, different pellet trajectories are considered for the ITER PIS, as shown in figure 2, and hence assumed in the simulations. There are two HFS lines, the upper and the lower HFS, as well as one LFS injection line. It should be noted that better fuelling efficiencies are expected for HFS injection lines than for the LFS one due to the outwards ∇B-drift. Therefore, HFS injection is expected to provide core fuelling, while LFS injection is projected for Edge Localized Mode (ELM) control. Indeed, a large ∇B drift is expected for LFS pellets, thus their residual core plasma fuelling would be rather low [57]. However, it is important to note that it might not be possible to reduce the residual core fuelling from LFS pellets by reducing their size, since such fuelling efficiency is a non-monotonic function of the pellet size [58].
The implication here is that ITER pellet fuelling efficiency will depend on plasmoid drift to achieve the necessary efficiency values. Thus, understanding all drift mechanisms is essential. Hence, a set of simulations for a range of pellet sizes and velocities have been carried out in this work using HPI2 and SMART. Deviations in the prediction of the pellet particle deposition profiles obtained with both codes can be interpreted as an indication of the level of uncertainty in the prediction to be expected due to differences in model assumptions, in particular for the plasmoid drift behavior. Taking into account that larger HFS pellets at maximum velocities have better fuelling efficiency, DT pellets with V p = 92 mm 3 , v p = 300 m s −1 are considered as nominal fuelling pellets. For pedestal width, similar assumptions to those in [57,59] are made, which predict an ETB width in the midplane ∆ ETB ∼ 9 cm for an ITER baseline scenario plasma (15 MA/5.3 T DT baseline scenario). In addition, simulations for the Half-field/Half-current or 7.5 MA/ 2.65 T scenario, at 7.5 MA/2.65 T, ∆ ETB ∼ 7 cm [59], and for the low-field 5 MA/1.8 T H Pre-Fusion Power Operation (PFPO) phase scenario [60,61] are performed, in order that the effect of background magnetic field can also be evaluated. It should be noted here that the local width of the ETB is different for each PIS injection line. In addition, due to the Shafranov shift, the outer ETB is smaller than the inner one, even in the midplane. The target plasma profiles, at the moments of pellet injection, are calculated by JINTRAC [62] with turbulent core transport being modeled by GLF-23 [63] or EDWM [64], neoclassical transport being calculated by NCLASS [65] and the edge transport enhancement due to ELMs being described by the continuous ELM model [66]. The central electron densities of these plasmas are 1.3 × 10 20 m −3 , 6.6 × 10 19 m −3 and 2.1 × 10 19 m −3 while the central electron temperatures are 15 keV, 10 keV and 13.5 keV, respectively (see figure 3). In these simulations, the effect of fast particles on pellet ablation is not taken into account. The reason for not including such an effect is that it has been found negligible due to a very low density of alpha particles and NB-heated ions in the plasma periphery. Moreover, the influence of the plasmoid drift in the final pellet particle deposition is studied by varying its effect from 100% of its predicted value to almost 0%, i.e. practically no drift is considered in calculations, which implies that particle deposition corresponds to the ablation profile. In order to perform such a scan on plasmoid drift, the drift is calculated using the normal procedure and then multiplied by a user-determined weighting factor that ranges from 0 to 1. This implies that no particular term is neglected in the calculation of the drift, as all terms are weighted by the same factor. Furthermore, the sensitivity to pellet size and velocity is evaluated by varying their nominal values. For instance, simulations with pellets that have 75%, 50% and 25% of the nominal pellet volume are carried out, while constant pellet velocities of 200 m s −1 and 500 m s −1 are also used as input.
From all the considered cases, predictions of the ablation and deposition profiles obtained with HPI2 and SMART for the representative case: a pellet of nominal size and velocity, i.e. V p = 92 mm 3 and 300 m s −1 , injected from the upper HFS into the Q = 10 baseline plasma scenario are shown in figures 4(a) and (b), respectively. Here, it is seen that, according to both HPI2 and SMART, the pellet penetrates to a depth (i.e. it is fully ablated) of ρ ∼ 0.95, i.e. to a depth of about 14 cm into the plasma along its trajectory path for the 15 MA/5.3 T scenario. However, much deeper particle deposition (i.e. final distribution of the ionized ablated particles of the pellet following plasmoid drift) is expected to be achieved due to the inwards plasmoid drift. Particularly, both SMART and HPI2 predict deposition profiles with maxima at ρ ∼ 0.89 and ρ ∼ 0.87, respectively. Additionally, in this figure, results for a pellet with the same parameters injected into the 7.5 MA/2.65 T scenario, i.e. 7.5 MA/2.65 T, ∆ ETB ∼7 cm, and into the 5 MA/1.8 T PFPO scenario are shown. For the former scenario, both codes predict deeper pellet penetrations, up to ρ ∼ 0.92. This could be expected since plasma temperatures in the 7.5 MA/2.65 T scenario are significantly lower than in the Baseline scenario (see figure 3). Regarding depositions, profiles are, to some extent, broader and with lower maxima. In addition, these maxima are shifted inwards, being the SMART prediction slightly more optimistic: its maximum is at ρ ∼ 0.78, while at ρ ∼ 0.82 for HPI2. The most optimistic predictions are obtained with HPI2 and SMART for the 5 MA/1.8 T scenario. Both codes predict a pellet penetration roughly similar to that for the 7.5 MA/2.65 T scenario (ρ ∼ 0.90-0.92; differences in the ablation particle density are caused by variations in dV/dρ and the amount of ablated particles because of different pellet particle densities for H vs. DT pellets). However, the deposition profile is broader and shifted even further inwards. Despite visible differences in details of the results obtained with HPI2 and SMART, the plasma domain where the largest fraction of the pellet material is predicted to become deposited is quite similar for all three scenarios considered and also the predicted trends as function of background plasma conditions appear to be in reasonably good agreement.
The fact that both codes apply quite different model approaches and approximations to describe the physics involved in the ablation and plasmoid drift processes but yield qualitatively comparable results for the pellet particle deposition profiles can be interpreted as an encouraging result that may provide increased confidence in the pellet source predictions for ITER. Nevertheless, it should be considered that significant uncertainties remain due to the fact that the models have been validated against plasmas in present day machines, with rather deep ablation profiles. For instance, it should be noted that, according to simulations in H-mode reactor size plasmas, a pellet is ablated within the pedestal. However, in present day experiments the ELM triggering by pellet starts when a pellet reaches half of the pedestal width. This is not taken into account in present simulations and could affect the final deposition and fuelling efficiency. Thus, the predictive capabilities of the pellet-simulation codes for the case of fusion reactors, such ITER or DEMO, requires further investigation. As starting point, low field, low density ITER scenarios could be considered. Such scenarios are planned for the initial PFPO phase of ITER. Therefore, the first H-mode plasmas achieved in the ITER PFPO phase might provide ideal and unique conditions for an in-depth validation of the models for the description of the plasmoid drift process in reactor relevant conditions.
To understand the differences in deposition profiles predicted by HPI2, which are not satisfactorily explained considering only the differences in ablation profiles, the average drift displacements calculated with HPI2 for several plasmoids along the pellet trajectories are plotted in figure 4(c)) for these same cases. It is observed that, in general, the total drift increases along the pellet trajectory due to the rising plasma temperature. Indeed, as a pellet penetrates the plasma, the increasing temperature enhances the ablation rate, and hence, because of the augmented cloud density, the initial pressure of the plasmoid grows. Such an increment of plasmoid pressure implies a larger difference between plasma and plasmoid pressures, which, in turn, raises the initial plasmoid drift. Also, some changes of tendency are predicted. For instance, there exists a difference in locations where plasmoids are heated and where they are finally deposited. This implies that, since the time scale for the balance of plasmoid and background plasma electron temperatures is much shorter than the time scale for plasmoid homogenization, the plasma electron temperature is reduced in rather close proximity to the location of plasmoid detachment also in case of a significant plasmoid drift displacement. Such temperature reduction leads, for an inward drift, to a more localized reduction of the temperature in front of the pellet, i.e. an enhancement of the pre-cooling effect [67]. The pre-cooling effect should reduce the ablation rate to values lower than would be expected for the target temperature profile, i.e. dN/dt ∝ Te 1.72 . Therefore, the initial plasmoid pressure, and hence, the plasmoid drift should also be lower than would be expected. This effect seems to be significant for ITER cases when a plasmoid heating close to the location of detachment is considered in the simulations (in agreement with the expected time scale for the heating), since it causes a clearly noticeable increase of the pellet penetration depth but a somewhat reduced plasmoid drift displacement. In addition to this general trend, i.e. the relationship between plasma temperature and plasmoid drift, a reduction of the drift may be observed due to the ECC effect in the vicinity of rational flux surfaces. This effect is particularly strong around ρ ∼ 0.93-0.95, depending on the scenario, where the drift shows a significant diminution. The reason for such a reduction is that external plasmoid self-reconnection lengths are shorter near rational flux surfaces, which causes an enhancement of the drift deceleration due to plasmoidexternal Pfirsch-Schlüter currents. When drifts calculated for both the 15 MA/5.3 T vs. 7.5 MA/2.65 T scenarios are compared, it is found that the drift is larger for the 7.5 MA/2.65 T scenario for plasmoids that have become detached at similar pellet locations, although the electron temperature is lower, as are the ablation rate and the initial pressure of the plasmoid. This difference, which causes the pellet particles to be deposited slightly further inwards for the 7.5 MA/2.65 T scenario, can be explained by a dependence of the dominant drift damping term that is due to the ECC effect on the square of the magnetic field. For the 7.5 MA/2.65 T scenario, this term is therefore ∼4 times smaller (neglecting differences in conductivity which also enter this term) as compared to the 15 MA/5.3 T scenario. In addition, the ∇B drift damping term due to the propagation of Alfvénic waves from the plasmoid, which appears before the external Pfirsch-Schlüter currents are formed, is also proportional to B 0 (note that B 0 is toroidal vacuum magnetic field at R 0 , defined as R 0 = (Rmax+R min ) 2 ). For that reason, one can expect plasmoid drift to increase at smaller toroidal fields. Indeed, the scaling formula for the average drift displacement given in [68], derived from a large set of simulations performed with HPI2, features a proportionality factor B 0 ∼(−0.35 to −0. 45) . This factor is in rough agreement with the obtained variation in plasmoid drift at locations with similar temperature environments when comparing the 7.5 MA/2.65 T case with the reference 15 MA/5.3 T plasma configuration. Therefore, the reduced drift damping efficiency in the 7.5 MA/2.65 T case slightly overcompensates the effect of reduced initial drift acceleration due to the lower temperature environment. Such larger drifts will account for the differences in deposition profile, i.e. the broadening of the profile and the inwards shift of its maximum. The same considerations also apply for the H-mode target plasma with application of 30 MW of ECRH in the 5 MA/1.8 T H PFPO phase scenario. Compared to the 15 MA/5.3 T and 7.5 MA/2.65 T scenario plasmas, the pellet particle deposition profile is predicted to be shifted significantly towards the core, the particle deposition profile barycenter reaching ρ bary,dep ∼ 0.8, even though the temperatures in the pedestal where the ablation takes place are lower than for 15 MA but comparable to the 7.5 MA plasma(T ped ∼ 3.5 keV as compared to T ped ∼ 5.5 keV and T ped ∼ 3.0 keV for the 15 MA/5.3 T and 7.5 MA/2.65 T scenarios, respectively). At these temperatures, the pellet penetration is slightly enhanced (compared to the 15 MA/5.3 T configuration), but the main contribution to the deeper deposition of the pellet particles in the 5 MA/1.8 T scenario configuration stems from an increased net plasmoid drift displacement. Again, a reduced drift acceleration caused by lower T ped in the ablation zone is overcompensated by a significantly reduced drift deceleration due to the ECC effect (factor ∼ 0.1 and ∼ 0.45 compared to the 15 MA/5.3 T and 7.5 MA/2.65 T scenarios because of the B 0 2 dependence in the ECC drift deceleration term).
To highlight the importance of plasmoid drift, a scan in drift is performed using HPI2 for the representative injection configuration in the baseline scenario. As mentioned above, all drift-related effects described in appendix A are considered in the calculation of the drift, and then weighted by the chosen factor. In this case, the factors used are 1 for the Full drift case, 0.5 for the Half drift case, and 0.1 for the <10%-drift case. Obtained ablation and deposition profiles are shown in figure 5. To properly evaluate the effect of the plasmoid drift on the deposition profile, it should be noted that pellet penetration may also be affected by the different magnitudes of considered drift due to the above-mentioned pre-cooling effect. To avoid such situation, in this scan, the relative change of the impact due to pre-cooling as function of the considered fraction of the calculated drift displacement has been neglected. Therefore, all differences in the deposition profile are directly related to the reduction of the considered drift. Focusing on the differences in deposition profiles, it is observed that, as expected, when a very small fraction of the calculated drift of the order of ∼<10% is taken into account in the plasmoid homogenization phase, the shape of the deposition profiles is similar to that for ablation (but slightly broadened), and, as the magnitude of the considered drift is increased, the deposition profiles get broader, and their maxima shift inwards.
In addition, to emphasize the importance of plasmoid drift as the mechanism for achieving the necessary deposition of the ablated particles, and hence the required electron density profile, HPI2 is utilized to compare all the foreseen injection directions. The obtained ablation and deposition profiles, as well as the predicted drifts, for the nominal pellet and the baseline scenario are plotted in figure 6(a)). As can be clearly observed therein, inwards drift is essential to achieve a deep deposition of the pellet material. Finally, the differences in plasmoid drift are emphasized in figure 6(b)), where the average drift displacement of several plasmoids detached from a pellet along its trajectory is plotted for these same cases. Besides the general trends previously described, it is apparent that larger total drift displacements are always predicted for injections from the upper HFS port than for the lower HFS. In contrast, the predicted effective drift for LFS is always significantly smaller than for both HFS cases. This is because, the plasmoid drift being outwards directed for LFS pellets, most plasmoids are predicted to reach the Last Closed Flux Surface (LCFS) before they completely slow down, and hence, they are lost. On the contrary, for both HFS cases, plasmoids are able to drift until either pressure equilibrium is reached or external and/or internal Pfirsch-Schlüter currents reconnect oppositely charged regions of the plasmoid. Such differences account for the different deposition profiles despite the similar ablation profiles. From the point of view of deeper fuelling, the upper HFS port is more beneficial than the lower HFS port, as the drift direction is better aligned with the normal to the flux surfaces for plasmoids with the barycenter being located close to the mid-plane (considering that only the flux surface-normal drift component leads to an inwards deposition of the plasmoid particles).
Moreover, HPI2 results for the particle deposition profiles from the scans in pellet mass and speed for the 15 MA/5.3 T scenario configuration are shown in figures 6(c) and (d) for the upper HFS case. It is clearly observed here that deeper deposition of pellet particles can be reached for larger pellet mass. This is partly caused by increased pellet penetration depth at larger pellet volumes. However, this effect is also amplified by larger initial plasmoid pressures in higher temperature environments, as those reached in interior regions, giving rise to improved deposition of the pellet particles towards the plasma core due to enhanced plasmoid drift displacement. The same mechanisms are responsible for improved deposition of the pellet particles at higher pellet injection velocities (increased pellet penetration and increased ∇B drift in interior regions). The variation in the particle deposition depth may have a sizeable impact on the expected fuelling efficiency, as the fraction of particles that are deposited inside the pedestal zone, which may be affected by ELMs, is predicted to increase substantially with increasing pellet mass or speed.

Wendelstein 7-X
The Wendelstein 7-X is a five-period, low-shear, driftoptimized stellarator, included in the HELIAS line, located at the Max-Planck-Institute für Plasmaphysik, Greifswald, Germany [69]. With an on-axis magnetic field of 2.5 T, it was designed to demonstrate that magnetic field optimization results in significantly better plasma performance, and hence, demonstrate the capability of the HELIAS-type stellarator as a possible alternative for a fusion power plant [52,70]. Its major radius, R 0 , is 5.5 m, and its averaged minor radius, <a>, is 0.55 m, which corresponds to a plasma volume of ∼30 m 3 [71]. Pellet injection in W7-X is an essential tool to achieve its scientific goal of long-duration, high-density plasmas (⩾10 20 m −3 ). During the second part of its initial operational phase, OP 1.2, a recommissioned blower-gun [27] was available for LFS and HFS injections [28,29,72]. For future operational phases, an optimized pellet injector will be available [29,73].
In this work, simulations of protium pellet injections have been carried out for one HFS (AEL41) and two LFS injection ports (AEK41-standard LFS-and AEE41-bean-shaped LFS). The plasma cross-sections for the three port sectors are shown in figure 7, together with nominal pellet trajectories. In addition, the plasma scenario, shown in figure 8, that corresponds to the maximum available ECR-heating in X2-mode (High Temperature-Profiles scenario) is considered (for other scenarios, see for instance [30]). For this case, the central electron density is 8 × 10 19 m −3 , while the central electron temperature is 7.5 keV. This scenario has been chosen for comparison with ITER, since it corresponds to the highest temperature expected to be attainable in W7-X, and hence, the closest to ITER  scenarios. It should be noted that experimental profiles are not used since the purpose of this work is not the validation of predictions (for this, see [29]). Instead, different drift scenarios, representative of the main operational target conditions of this machine, are assessed to enable a complete comparison with ITER cases. Regarding pellet parameters, cylindrical protium pellets, with volumes ranging from 9 mm 3 to 74 mm 3 , injected at velocities from 200 m s −1 -1000 m s −1 are considered in the simulations. It should also be noted that the larger pellet sizes for W-7X are comparable to those in ITER, while the W-7X plasma volume and particle content are typically about 30 times smaller. This implies that the pre-cooling effects in W-7X is more sizeable than those in ITER.
As for the ITER case, the effect of the plasmoid drift on particle deposition is investigated by varying its theoretical value between 100% and a few percent. Again, all drift-related effects included in HPI2 are considered in the calculation of the drift, and then weighted by the chosen factor. The considered cases are the Full drift case, the Half drift case, and the <10% -Drift case. Of all the possible combinations of pellet and plasma parameters, the case of a pellet of V p = 42 mm 3 and v p = 200 m s −1 , injected into the High Temperature-Profiles scenario, is selected as a representative example here. The ablation and deposition profiles, obtained from the drift scan of such a case, are shown in figure 9, for the HFS case. In general, the pellets penetrate to less than 10 cm into the plasma (5-9 cm, depending on the injection port and the magnitude of drift considered in the calculations), and reach ρ ∼ 0.7-0.8 along the trajectory path. It should be noted that the minor differences in ablation profiles are due to the pre-cooling effect. For instance, for the Full drift case, pellet penetration increases by ∼1% due to pre-cooling. Similarly, for the Half drift and It should be noted that fuelling efficiency for the bean shaped LFS is significantly low, but non-zero (it is 0.7 %, in contrast to 18% and 98% for the standard LFS and the HFS ports respectively). (b) Total drift for several plasmoids, detached from the pellet at different ρ coordinates along its injection path, for the same scan in pellet injection location (bean-shaped LFS in dashed-dotted green, standard LFS in dotted red and HFS in continuous blue). The plasmoid drift until the LCFS only is taken into account in the case of LFS injection. There is one plasmoid for each of the equally spaced steps (2 mm) along the pellet trajectory for which pellet ablation is calculated. Plasma target profiles are shown in figure 8.

<10%-Drift cases, penetration increases by ∼4% and ∼5%,
respectively. In addition, like the tokamak case, the magnitude of the considered drift leads to very differing pellet particle depositions, and hence, to different deposition profiles. Indeed, as expected, the deposition profile is equivalent to the ablation profile when a small fraction, of the order of a few percent, of the calculated drift is included, and it gets broader, while the maximum is inwardly shifted, as the amount of considered drift is increased. It is worth noting that, while for the Full drift case the changes in the deposition profile due to pre-cooling are negligible, the differences for the other two cases can be very significant. The reason is that, in the Full drift case, the slightly deeper pellet penetration due to precooling is counterbalanced by a smaller drift (∼0.03 m for each plasmoid). However, for the other two cases, the already shorter drift is further reduced for each plasmoid by ∼0.3 m and ∼0.002 m, respectively, which implies that the plasma is cooled closer to the pellet location. This makes the reduction of the ablation rate, and thus, the enhancement of the pellet penetration, more effective, i.e. for the Half and <10%-Drift cases the reduction of the drift is not only favorable from the deposition point of view but leads to a significant increase in pellet penetration.
In order to emphasize the effect of injection location, the ablation and deposition profiles obtained when full drift is considered are plotted together for the three injection ports in figure 10(a)). In this figure, it can be observed that the deepest material deposition, as for ITER, is obtained for HFS injection, since for LFS injections, the plasmoid drift reduces pellet material deposition. However, the minor plasma radius flux surface coordinate gradient ∇ρ is increased along the pellet injection path allowing for deeper pellet penetration for LFS injection compared to ITER. Moreover, different post-injection profiles are obtained for the two LFS injection ports, with the standard LFS being the more beneficial of the two [30]. For these same cases, the average drift per plasmoid particle in normalized minor radius flux surface units is plotted in figure 10(b)). Here, as noted for ITER simulations, it is observed that the total drift increases with pellet penetration, due to the increment of the plasma temperature, which increases the initial plasmoid drift acceleration. Again, local reductions, in the vicinity of rational flux surfaces, are found, due to the shorter self-reconnection length corresponding to these surfaces. However, their effect is considerably less pronounced than that in the tokamak case, since the effect of external currents in the drift damping process is much smaller than the one produced by plasmoid-internal currents. In addition, these local reductions are less frequent in W7-X than in ITER due to its different q-profile, for which only higherorder rational flux surfaces exist, opposite to tokamaks (see figure 11).
After that, to complete the analysis of the effect of confining magnetic field geometry on plasmoid drift for stellarators, particularly for W7-X, a scan in plasma beta, β, i.e. the ratio of the plasma pressure to the magnetic pressure, is performed. Such a scan is carried out to assess the expected pellet ablation and particle deposition behavior in W7-X experiments for varying β, due to the associated modification of the magnetic field. For that, three versions of the standard magnetic configurations with different β, namely High β standard case (β = 0.04), Medium β standard case (β = 0.02) and Low β standard case (β = 0.0032) are studied. It should be noted that these are not different magnetic configurations per se, such as the high mirror or the high iota, but the same configuration, the standard, including the changes associated to the increment in β. Their corresponding density and temperature  (a) Electron density corresponding to High β-Profiles (continuous green) and to Low β-Profiles (dotted magenta) in W7-X and (b) electron and ion temperatures corresponding to these same scenarios, being the electron temperature of the High β-Profiles case represented by continuous green, while by dotted green the ion temperature. In addition, the electron temperature from the Low β-Profiles is plotted in dashed magenta, whereas the ion temperature is in dash-dotted magenta.
profiles are also considered in the simulations. For the Medium β case, the same plasma scenarios shown in figure 8, i.e. the High Temperature-Profiles (hereinafter referred as the Medium β-Profiles scenario) and the High Density-Profiles scenarios, are used. In contrast, for High β and Low β cases, the density and temperatures shown in figure 12 are used as input for these simulations. Hereinafter, these scenarios are referred as the High β-Profiles scenario and the Low β-Profiles scenario, respectively.
This part of the work has been divided into two, since it is necessary to isolate the true impact of the changes in the magnetic configuration, such as the magnetic well or the shift of the magnetic axis, from the increment of plasma pressure, given that both affect the ablation rate and the plasmoid drift. First, a scan, varying the magnetic configuration, while keeping fixed background plasma assumptions, although such profiles might only be consistent with a specific β magnetic configuration, is executed to evaluate the direct impact of the change in magnetic field configuration with varying β on the plasmoid drift behavior. Afterwards, a second set of simulations with varying plasma profiles but fixed magnetic configuration is performed. In addition, regarding injection ports, pellets are foreseen to be injected from the same injection ports used in the first part of the study (see figure 7). Moreover, in order to complete the analysis, pellet velocities ranging between 150 m s −1 and 1000 m s −1 are considered in the simulations, whereas a range of pellet diameters and lengths is also included in this study. It should be noted that no changes in the magnetic field It should be noted that fuelling efficiency for the High β Configuration is significantly low, but non-zero (it is 0.7%, in contrast to 18% and 74% for the Medium and High β Configurations respectively) (b) Total drift for several plasmoids, detached from the pellet at different ρ coordinates along its injection path, for the same β-scan (Low β in continuous blue, Medium β in dotted red and High β in dashed-dotted green). There is one plasmoid for each of the equally spaced steps (2 mm) along the pellet trajectory for which pellet ablation is calculated. Only the plasmoid drift inside the confined region is taken into account 9 . configuration are taken into account as the increase in density associated with a pellet injection is assumed to be an adiabatic process 10 .
Starting with the scan in magnetic configuration, the chosen scenario is the Medium β-Profiles scenario (figure 8). Ablation and deposition profiles obtained for the representative case, i.e. a H pellet with a volume of V p = 42 mm 3 , injected at 200 m s −1 , into the standard LFS port, are shown in figure 13(a)). Here, it is observed that, if plasma temperature is removed from the possible affecting variables, a clear improvement in particle deposition with increasing β magnetic configurations is noticeable. In addition, pellet penetration is expected to improve slightly for higher β configurations due to the increment of the outwards displacement of the center of flux surfaces, or Shafranov shift, with plasma β. It should be noted here that, although plasma profiles (as function of ρ) are fixed for this scan, the changes in magnetic configuration associated to an increment of plasma pressure, such as the Shafranov shift, are included in each of the magnetic configurations considered in the scan. Nonetheless, temperature plays a minor role in the results, since for higher betas the deposition is deeper, i.e. during the homogenization phase, plasmoid drift eventually becomes inwards directed, and hence the possibility of pre-cooling arises, enhancing pellet penetration. This can be clearly observed in figure 13(a)), where it is seen that pellet penetration significantly increases with β, and hence, pellet deposition becomes deeper, this being accompanied by a higher fuelling efficiency, due to the different effective drifts ( figure 13(b)).
Therefore, the variation in pellet ablation with magnetic configuration can be explained by the differences in particle deposition. In turn, deposition profiles are explained by magnetic configuration variations since they lead to different plasmoid movements. Indeed, plasmoid drift behavior changes significantly with β. To explain such differences, it must be considered that, when central β increases, the magnetic well is enhanced due to diamagnetic flux expulsion. This leads to an enhanced inwards particle drift, compared to low β cases, since the additional ∇B component points away from the plasma center. This can be clearly observed in figure 14, where trajectories of a plasmoid barycenter during the whole homogenization time, for plasmoids originated at different pellet positions, are displayed. In this figure, the following is observed: for the Low β configuration case, plasmoids tend to drift mostly outwards, without any change of its direction from the radial position where they are detached to almost the LCFS. On the contrary, for the High β configuration, plasmoid direction changes several times during the whole homogenization time, being mainly inwards directed. Meanwhile, for the Medium β configuration case, plasmoid trajectories show a combination of both trends, being outwards directed for plasmoids detached in the plasma edge region, while principally inwards with changes in the direction for plasmoids detached in inner regions of the plasma. It should be noted that for this case the trajectories of two different plasmoids are shown, each of them detached from the pellet at regions of different drift trends. Therefore, the pre-cooling effect is only significant for the High β case, for which pellet penetration is increased by ∼7%, as it is the only case where plasmoids drift inwards even for LFS injection, i.e. the plasma cools in the direction of the pellet. On the contrary, for the Medium β and Low β cases, penetration is only increased by ∼0.3% and ∼2%, respectively, due to the pre-cooling. The differences in plasmoid drifts, and thus in deposition profiles, caused by the pre-cooling effect are negligible.
Results of the scan in background plasma profiles, for a fixed magnetic configuration, the Medium β standard configuration, are presented in figure 15 to highlight the importance of isolating the variation of the magnetic configuration from, particularly, the increase of electron plasma temperature. In figure 15(a)), ablation and deposition profiles are shown, while the average drift for the same cases is presented in figure 15(b)). Since magnetic configuration and pellet parameters are fixed, all differences can be attributed to the different plasma densities and temperatures. Indeed, since it is known that the strongest dependence of ablation rate is on background electron temperature, the lowest ablation rate, and hence, deepest pellet penetration is obtained for the Low β-Profiles case, while the opposite is predicted for the High β-Profiles scenario. However, it is observed that the differences in ablation profile and pellet penetration between the Medium β-Profiles and the High β-Profiles cases are not as significant as the variances with the Low β-Profiles scenario. Such predictions for pellet penetration result in deposition profiles that completely differ between the three cases. For instance, for the High β-Profiles scenario, the number of particles deposition inside the confined region is relatively negligible, with a maximum at ρ = 1; while for the Low β-Profiles scenario, the deposition profile is shifted significantly towards the center (maximum at ρ ∼ 0.5). The deposition profile for the Medium β-Profiles scenario presents characteristics that are in-between the other two. To account for this, the geometry of the magnetic field corresponding to the regions traversed by a pellet must be taken into account. Considering this, total plasmoid drifts for the three cases are essentially equal along the common pellet path with small differences in the drift magnitude due to the different plasma temperatures (see figure 15(b)). Main differences in deposition profiles are attributed to the deeper pellet penetration achieved in lower temperature scenarios, since a pellet is able to travel inside regions of different curvature radii, some of them beneficial in terms of drift direction, before being fully ablated. This results in different plasmoid effective drifts, and hence, in radically different particle deposition profiles.
In summary, for the studied cases, plasmoid drift varies moderately with β in the initial stages, when the plasmoid longitudinal extension is still small, as observed in figure 16, where the radial component of the effective inverse curvature radius, averaged along different plasmoid parallel lengths, is found (note: plasmoid drift acceleration in radial direction is proportional to this quantity). Such component of the effective inverse curvature radius vector, ⃗ r inv , averaged over plasmoid extension, is defined as [26]: The quantity expressed in equation (7) corresponds, for small plasmoid extensions, to the 1/R curv projected to the flux surface normal in tokamaks. However, as the plasmoid expands, the averaged inverse curvature radius is appreciably altered. This results in significantly different plasmoid accelerations, particularly at the plasma periphery and in the interior regions, near the plasma center, at least for the β interval that has been tested. These observations can be explained by considering that, for a magnetic well structure, the long-term drift behavior may be favorable, while unfavorable for a magnetic hill structure, regardless of the injection position [26]. For W7-X, the magnetic field configuration presents a small well structure for low β that becomes deeper as β increases. Therefore, for low β configurations, the plasmoid drift is predicted to be more outward directed, in contrast to higher β configurations, for which the drift tends to be more inwards directed. These should account for the differences between previous HPI2 predictions for HFS injection in W7-X and results obtained in the last experimental campaign, OP1.2 [29,30]. In addition, this implies that, for deep pellet penetrations (deeper than ρ ∼ 0.5), the deposition profiles should be roughly similar, less dependent on the magnetic configuration. This is because the deeper pellet penetration at low β is compensated by an increased outward directed drift, while for higher β, a shallower penetration is counterbalanced by a more inward directed drift. It should be noted that fuelling efficiency for the High β Profiles is significantly reduced, but non-zero (it is 0.6 %, in contrast to 18% and 97% for the Low and Medium β Profiles respectively) (b) Total drift for several plasmoids, detached from the pellet at different ρ coordinates along its injection path, for the same plasma profiles scan (Low β-Profiles in continuous blue, Medium β-Profiles in dotted red and High β-Profiles in dashed-dotted green). Only the plasmoid drift inside the confined region is taken into account. There is one plasmoid for each of the equally spaced steps (2 mm) along the pellet trajectory for which pellet ablation is calculated. Plasma target profiles are shown in . Figure 8 (the Medium β Profile scenario is referred here as the High Temperature-Profiles case) and figure 12.  In addition, to illustrate the effect of the changes in magnetic configuration associated with an increasing β described above, an additional scan has been run for the standard LFS injection case. The aim of this supplementary study is to determine the minimum injection velocity necessary to achieve a deposition profile with maximum value at ρ = 0.8 for a pellet size of V p = 42 mm 3 , and for the different combinations of magnetic and plasma configurations. Results are presented in figure 17, where it can be observed that the variations in the magnetic configuration due to the increment of β indeed have a large impact on the deposition behavior.
Finally, for completeness, results of a scan in plasma β using consistent background plasma profiles are presented in figure 18. In figure 18(a)), ablation and deposition profiles are shown, while the average drift for the same cases is presented in figure 18(b)). In this case, differences are attributed to a combination of the different plasma densities and temperatures, and the differences in the magnetic configuration. Similar to the results for the scan of plasma scenarios, the lowest ablation rate, and hence, deepest pellet penetration is obtained for the Low β case, while the opposite is predicted for the High β case. Moreover, it is observed again that the differences in ablation profile and pellet penetration between the Medium β and the High β cases are not as significant as the variances with the Low β case. However, predictions for deposition profiles differ from the scan in plasma scenario. For instance, for the High β case, the number of particles deposited inside the confined region is not negligible and has a maximum at ρ = ∼ 0.77; being very similar to the deposition profile for the Medium β case (its maximum is at ρ = ∼ 0.75).
The deposition profile for the Low β case is shifted significantly towards the center (maximum at ρ ∼ 0.6). Furthermore, total plasmoid drifts for the three cases are essentially equal along the common pellet path with small differences in the drift magnitude due to the different plasma temperatures and magnetic configurations (see figure 18(b)). Therefore, differences in deposition profiles are attributed to a combination of the deeper pellet penetration achieved in lower temperature scenarios, and the different plasmoid drift trends observed in figure 14. Besides this, it is clearly seen that the drift for the Low β case with consistent plasma profiles and magnetic configurations presents larger drifts than the case of Low β scenario with Medium β configuration. All these results highlight the role played by changes in the magnetic configuration associated with an increment in plasma β.

Discussion: the influence of device configuration
The different magnetic configurations inherent to tokamaks and stellarators, and here for ITER and W7-X, respectively, are in principle expected to result in differences in fuelling efficiency, since the particularity of each device type may lead to different drift characteristics and dependencies, in particular with respect to the relative importance of the drift-damping mechanisms. The main differences are summarized in table 1.
For both ITER and W7-X, the injection of pellets from the HFS is predicted to be advantageous for improved core fuelling. However, while a major difference is predicted for the fuelling efficiency with HFS vs. LFS injection for ITER, HFS injection is predicted to be moderately more advantageous than LFS injection for W7-X as compared to ITER. In low β configurations of W7-X, the difference in total fuelling efficiencies obtained for HFS vs. LFS injection may be within measurement uncertainties (although deposition profiles are more peaked for HFS; see table 2 and figure 19), as the plasmoid drift displacement is predicted to be very small, not only due to a low temperature background, but also because of the effective inverse curvature radius in the drift acceleration term approaching zero for extended plasmoid elongation, as shown in figure 16(d)). Similarly for high β plasmas in W7-X, the less efficient fuelling by LFS can be compensated by injection of the pellets at a higher velocity, which is possible because of the simpler injection geometry on the LFS. Meaning that, in medium or high β configurations, the plasmoid drift displacement may become significant. However, an inward directed drift acceleration of the same size is predicted for both HFS and LFS injection in the later plasmoid homogenization phase, when the longitudinal plasmoid extension has become sizeable. This is because the |∇ρ| oriented component of the effective inverse curvature radius becomes negative and averages to similar values along flux surfaces, as illustrated in figures 16(e) and (f ). Differences in the drift acceleration for HFS vs. LFS injection may only be present in the early plasmoid homogenization phase as shown in figures 16(b) and (c). As a result, HFS injection may provide a deposition profile barycenter that is located further inwards, but its advantage for deep fuelling compared to LFS injection may be less . Only the plasmoid drift inside the confined region is taken into account. There is one plasmoid for each of the equally spaced steps (2 mm) along the pellet trajectory for which ablation is calculated. Plasma target profiles are shown in . Figure 8 (the Medium β Profile scenario is referred here as the high temperature-profiles case) and figure 12. Table 1. Comparison of device configuration specific aspects and dependencies of the pellet ablation and deposition processes in ITER vs. W7-X in HPI2 code simulations. The quantities eρ and R eff correspond to the unit vector in ρ direction and to the effective curvature radius, respectively, while a 0 is the plasma minor radius, defined as (a 0 = Rmax−R min 2 )..

ITER
W7-X   evident than in the case of ITER (see figure 20). In practice, the benefit of deeper fuelling due to a more inwards directed plasmoid drift for HFS injected pellets may become compensated by enhanced technical challenges. These are related to the need for a curved and lengthened guide tube for the pellet injector, which will limit the injection speed and mass. Therefore, larger pellets injected at higher speed from the LFS, may provide deeper pellet particle deposition due to enhanced pellet penetration, compared to small-sized pellets that need to be injected at reduced speed from the HFS (see figure 19). One may also need to consider that deeper deposition could also be reached for pellets injected from the standard LFS trajectory, as shown in figure 7(b)), as compared to pellets injected from the HFS trajectory, displayed in figure 7(a)), due to a substantially higher |∇ρ| along the pellet injection trajectory, which may be even further increased for LFS injection at high β due to the Shafranov shift. For stellarators, plasmoid-internal charge reconnection lengths are shorter than for tokamaks, as they scale with q. On the other hand, the effect from plasmoid-external charge reconnection may be small, since the rational q surfaces present in the plasma are of relatively low order. Therefore, in stellarators, the ICC effect is expected to dominate the drift damping process, while the ECC effect may only be relevant as a second order correction in relation to the ICC effect. On the contrary, for tokamaks, with longer plasmoid-internal charge reconnection lengths and higher order rational q surfaces, the ECC effect is predicted to be the dominant plasmoiddrift damping mechanism, while the ICC effect is only significant for large plasmoid extensions, i.e. in the late phase of the homogenization process, when the plasmoid pressure is already very low, and hence, the drift velocity has significantly decreased. To illustrate the difference in the importance of the ICC vs. ECC effects for the W7-X vs. ITER configurations under consideration, estimates have been made for some terms that are representative for the size of these effects for these configurations. For this, a few simplified assumptions have been made; in particular, for the radius of the cross-section of the plasmoid (prescription of a constant of 2 cm), the longitudinal expansion speed (taken to be the sound speed with a plasmoid temperature being set to 50% of the background temperature), the plasma geometry (circular cross-section) and the effective charge of the plasma (Z eff = 1.7). The time required for the first external re-connection of regions with opposite Figure 21. Comparison of the time required for the first external re-connection of regions with opposite charges of the plasmoid (τ con,ECC , dash-dotted green lines, representative for the time required for the ECC effect to set in), the time required for plasmoid-internal re-connection to become effective (τ con,ICC , dotted red lines) and a proportionality factor that is included in the drift damping term due to the ECC effect in the drift equation (f ECC , continuous blue lines), for the outer region of the confined plasma in a) the 15 MA/5.3 T ITER baseline scenario and (b) the medium β W7-X scenario configurations, calculated with simplified plasma geometry and plasmoid assumptions (fixed perpendicular plasmoid extension of 2 cm, longitudinal plasmoid expansion speed taken to be equal to the sound speed at 50% of the background plasma temperature Zeff = 1.7). In (c) the ratio of ITER and W7-X ECC factors is shown in green dots on the left axis, while on the right axis the ratio τ ICC /τ ECC for ITER (dashed-dot purple line) and W7-X (continuous blue line) is found. charges of the plasmoid, τ con,ECC , which indicates the time required for the drift damping due to the ECC effect to set in the plasmoid homogenization process, has been calculated. Together with this, the time τ con,ICC required for the longitudinal plasmoid extension to reach a value q·π·R 0 , which is required for internal re-connections to become effective has been computed. The results for τ con,ECC and τ con,ICC are shown in figure 21 for the plasma periphery for the ITER 15 MA/5.3 T scenario and the W7-X Medium β-Profiles scenario configurations. If τ con,ICC > τ con,ECC , the ECC effect is more important, since the plasmoid drift velocity may be reduced mainly by external short-circuit currents before the ICC effect becomes active. This is the case for ITER in the plasma edge region where the plasmoid homogenization processes take place. If τ con,ECC > τ con,ICC , the internal Pfirsch-Schlüter currents appear before the external ones are established, which is the case for the W7-X configuration. In addition to τ con,ECC and τ con,ICC , a proportionality factor f ECC is shown in figure 21. It is included in the drift damping term for the ECC effect in the drift equation and describes the strength of the drift deceleration caused by external Pfirsch-Schlüter currents for a given plasmoid drift velocity (neglecting changes in the effective mass and cross-section of the plasmoid). f ECC is found to be significantly smaller (by a factor >∼5) for the W7-X configuration in the outer part of the plasma, which further emphasizes the weakness of the ECC effect (and relative importance of the ICC effect) in W7-X conditions and the presence of opposite relations in the case of ITER, where the ICC effect plays a subdominant role. It is worth noting that, due to larger plasma vs. plasmoid cross-sections and an increased longitudinal plasmoid expansion speed in a high temperature background, the ICC effect is more noticeable in relation to the ECC effect in ITER as compared to present-day large-size tokamaks, such as JET, or mid-size tokamaks such as ASDEX-Upgrade.
Finally, simulations presented here show that relatively larger net drift displacements are expected for W7-X due to a higher averaged inverse curvature radius of its magnetic field lines, which, for stellarators, scale with an average between the inverse of the major and minor radius rather than with the inverse of the major radius, as is typical for tokamaks (cf figures 16 and 20, which contains the contour plot of the ρ component of the effective inverse curvature radius for the ITER 15 MA/5.3 T configuration). The ∇B and curvature drift should thus be relatively larger, and, due to increased |∇ρ| because of the smaller machine dimensions (in terms of a 0 ), lead to a significantly enhanced drift acceleration of plasmoids in terms of ρ coordinates. This is indeed the case; however, shorter plasmoid-internal charge reconnection lengths result in the ICC effect being more effective than in a tokamak. In addition, since the drift acceleration term is also dependent on the plasmoid pressure, which strongly depends on the ablation rate and thus on the background electron temperature, the reduction in drift acceleration due to a higher curvature radius in ITER may be partly compensated by an increase in drift acceleration due to higher T e in the pellet ablation zone (even though pellet ablation is much more peripheral in ITER). This implies that the net drift velocity in the plasmoid homogenization phase may be comparable or even smaller in the case of W7-X. While the plasmoid drift may be helpful to achieve deeper deposition of pellet particles, such that a fuelling efficiency of 100% may be approached for HFS injection in ITER and both LFS and HFS injection in the W7-X medium and high β configurations (for medium β configurations see [30]), it should be noted that deep core fuelling by pellets (i.e. with a substantial fraction of pellet particles penetrating beyond ρ ∼0.5) may only be envisaged for W7-X low β scenarios because of the low plasma temperature. It should also be considered here that any of phenomena occurring after or due to pellet injection, such as ELM triggering, is neglected.

Summary and conclusions
In this section, the principal findings of this work are summarized first, starting with ITER, as the chosen tokamak, and continuing with W7-X, as the representative stellarator, before completing it with a comparison of results. Finally, the main conclusions derived from this study are presented below in an attempt to shed light on magnetic configuration dependent characteristics from the perspective of efficient pellet fuelling.
Several results can be highlighted from HPI2 and SMART simulations for ITER. First, very peripheral ablation profiles are obtained with pellet ablation taking place entirely within the pedestal region due the presence of large pedestal temperature gradients. For the 7.5 MA/2.65 T scenarios and the 5 MA/1.8 T PFPO scenario the pellet penetrates slightly more deeply into the plasma, as the ablation rate is reduced due to lower pedestal temperatures. In contrast, it is found that the plasmoid drift is increased for the 7.5 MA/2.65 T scenario and the 5 MA/1.8 T PFPO scenario configurations, although the electron temperature is lower as are the ablation rate and the initial plasmoid pressure, which is mainly due to a weaker ECC effect that scales with the square of B 0 . Larger drifts will account for the different deposition profiles: slightly broader and shifted inwards for the 7.5 MA/2.65 T scenario, and even further shifted inwards for the 5 MA/1.8 T PFPO scenario. Moreover, the total drift increases along the pellet trajectory due to the rising plasma temperature, with local variations close to rational flux surfaces, owing to shorter external plasmoid self-reconnection lengths near such regions. SMART predictions show visible differences in both ablation and deposition profiles for the three scenarios, but following an equivalent trend to HPI2 predictions, i.e. ablation is lower, and the pellet penetrates deeper into the plasma for scenarios with lower plasma edge temperature, and the plasmoid drift is enhanced with deeper pellet penetration to higher background temperatures and with reduced B 0 . Focusing on the differences in deposition profiles, it is found that, when only a small fraction of the predicted drift is taken into account, the shape of the deposition profiles is, as expected, similar to that of ablation profiles, and, as the magnitude of the considered drift increases, deposition profiles become broader and their maxima shift, inwards for HFS and outwards for LFS. Besides the general trends for plasmoid drift, larger drifts are always predicted for injection from the upper HFS, and smaller for the LFS. Finally, for the scan in injection location, ablation profiles are similar for the three foreseen injection ports, while deposition profiles differ substantially. Despite very peripheral pellet penetration limited to the ETB zone, significant particle deposition is predicted to occur in the core plasma up to ρ ∼ 0.80 for HFS injection depending on injection conditions. For LFS injection, the fuelling efficiency is close to zero with almost all pellet particles being predicted to be expelled from the confined region due to a strong outward directed plasmoid drift. Deeper deposition of the pellet particles can be reached for larger pellet mass and/or increased pellet speed. This is caused by a deeper penetration of the pellet itself, but also by larger initial plasmoid pressures in higher temperature environments. These temperatures are reached in interior regions and give rise to improved deposition of the pellet particles towards the plasma core due to enhanced plasmoid drift displacement. This may be important to maximize the fuelling efficiency relevant fraction of particles deposited inside the ETB or the ELM-affected zone.
Similarly, some results from W7-X simulations should be highlighted. Starting with the drift scan, ablation profiles vary slightly with the magnitude of the considered drift as a consequence of pre-cooling, between ∼1% for the Full drift and ∼5% for the 0.1%-Drift cases. In addition, just as with ITER results, deposition profiles are equivalent to ablation profiles when only a small fraction of the predicted drift is included, and they become broader, while the maximum is shifted, as the amount of considered drift is increased. Regarding the scan in injection direction, deepest depositions may be obtained for HFS since plasmoid drifts are mainly inwards, but inward directed drifts may also occur for pellets injected from the LFS in the later phase of plasmoid homogenization. Moreover, the total drift also increases with pellet penetration for W7-X, because of higher temperatures in the core. Turning to the scan in plasma β, for the variation of the magnetic configuration, it must be emphasized that, clear improvement in pellet particle deposition towards the core with increasing β is noticeable. This implies that, for the studied cases, the plasmoid drift varies moderately with β in the initial stages, when the plasmoid longitudinal extension is still small. However, as the plasmoid expands, the averaged inverse curvature radius is appreciably altered. This results in significantly different plasmoid drift behavior, particularly at the plasma periphery, with a more inward directed drift displacement occurring at higher β for both HFS and LFS injections. These observations can be explained by considering that, for a magnetic well structure, the long-term drift behavior may be favorable, while unfavorable for a magnetic hill structure, regardless of the injection position. Therefore, for low β configurations, the plasmoid drift is predicted to be more outward directed, in contrast to higher β configurations, for which the drift tends to be more inwards directed.
Having summarized the main finding for each device, the comparison of the drift characteristics in the ITER tokamak and the W7-X stellarator machines are now recapitulated. Indeed, some differences in plasmoid drift and, hence, in particle deposition profile are present. This would be expected since the ICC effect is the dominating drift damping process for stellarators, while ECC typically dominates for tokamaks. In addition, larger drifts are in principle expected for W7-X due to the smaller curvature radius of its magnetic field lines. These drifts may be even larger in terms of ρ due to the difference in machine dimensions. However, shorter plasmoidinternal charge reconnection lengths result in the ICC effect being more effective than in a tokamak. Moreover, the drift direction may change sign in a stellarator during the plasmoid homogenization process. Furthermore, lower temperatures are achieved in W7-X for the considered scenarios as compared to ITER in the periphery where the pellets become ablated, giving rise to lower plasmoid pressures and reduced drift acceleration. These effects may compensate the enhancement in the plasmoid drift due to smaller curvature radii and a more favorable |∇ρ|. Therefore, one of the main conclusions is that the net average plasmoid drift displacement may be comparable to ITER or even smaller for pellets injected in W7-X. HFS injection is predicted to be highly advantageous in order to maximize the core deposition of pellet particles in ITER. It may also be advantageous for enhanced core deposition of pellet particles in medium to high β environments in W7-X, though a more inwards directed plasmoid drift may also be expected for LFS injection in these conditions. In addition, LFS injection may benefit from a higher accessible pellet injection speed and pellet size in addition to a more favorable |∇ρ| that may counterbalance the advantage of HFS injection in terms of the achievable fuelling efficiency. While plasmoid drift may be helpful to achieve deeper deposition of pellet particles such that a fuelling efficiency of 100% may be approached in both devices, deep core fuelling by pellets may only be envisaged for W7-X low β scenarios. However, as mentioned above, these values of the fuelling efficiency, close to 100%, are without taking into account pellet material expelled by ELMs.
In summary, it can be concluded from these results that, at least for current device sizes and plasma scenarios, and despite the sizeable differences in the relative importance of the mechanisms of plasmoid drift acceleration and deceleration in play, there may be means for the optimization of the pellet injection configuration in both ITER and W7-X. This way, it could be possible to maximize the deposition of pellet particles towards the core and the fuelling efficiency in a way that allows for efficient but edge-localized fuelling of the core plasma by pellets. Not only that, but the study presented here may also be the basis to prepare well benchmark tools that serve for reactor optimization. In relation to this, it must be noted that pellets can be considered as multitasking tools. Indeed, they are used not only for core fuelling, but also as a versatile tool for mode control by triggering or performance enhancement and radiative cooling by supplying seeding gas [74,75]. As a final conclusion, it must be highlighted that well benchmarked tools, which could be derived from those presented here, will be necessary to optimize and integrate all pellet associated tasks, both related to fuelling and control, in a fusion reactor.
Here, m i and n i are the plasmoid ion mass and density, respectively, while B ∥ is the magnetic field in the parallel direction, and µ 0 is the vacuum permeability. In addition, v A is the Alfvén speed: where n ∞ is the background plasma density, and B local is the local value of the magnetic field. 3. A drift-damping term produced by short-circuiting of internal currents, referred as the ICC effect, as explained in section 2 [16,17]. This effect originates from the fact that, as magnetic field lines rotate around the plasmoid center, the charge accumulation on the plasmoid surface varies, or even changes sign, along a magnetic field line. This charge accumulation is then equilibrated by the plasmoid-internal short-circuit current. To account for this effect, it is sufficient to use the expressions in equation (6), instead of (4). 4. A drift-damping term produced by short-circuit of external currents, the so-called ECC effect, also introduced in section 2 [18]. This effect takes place when regions of opposite polarization become connected with each other, at a time larger than the time of first connection, con ≈ πR0m vA , which corresponds to a plasmoid reconnection length of L con ≈ 2πR 0 m. The expression that accounts for this effect is: Here, M 0,Lcon2 is the increased inertial mass, which is used, instead of the plasmoid mass, M 0 = L a L b Z 0 n 0 m i , to account for the momentum transferred from the plasmoid to the background plasma [31,77]: In addition, L a and L b are the plasmoid dimensions, as defined in figure 22, while L Jϕ is the size of the current channel [77]. Moreover, L conH and L conA are the harmonic and arithmetic average connection lengths, respectively. P con and P con2 are the fraction of once and twice reconnected charges, respectively. Finally, σ ∥∞ is the external plasma conductivity: In this expression, q e and m e are the electron charge and mass, respectively, while τ e∞ is the electron collision time (ln Λ is the Coulomb logarithm): It should be noted that the Alfvén-wave effect only applies to the fraction of non-externally reconnected charges. In addition, the ECC effect depends on the poloidal position of the plasmoid, since the distribution of charges inside the plasmoid can be asymmetric, and, hence, the fraction of reconnected charges can decrease. The fractions of reconnected charges as a function of the angles between the drift direction and the flux surface tangent, α [31,77], are P con (α) ∼ P con (α = 0) cos 2 (α) (A8) P con2 (α) ∼ P con2 (α = 0) + P con (α = 0) 1 − cos 2 (α) .
Considering all these terms, the two components of the drift equation become: As explained previously, the first term in both expressions corresponds to the driving-drift term. In addition, it includes the driving term, due to plasmoid parallel expansion and the ICC effect. The second term is related to the Alfvén wave damping effect, while the third one is the ECC effect.

Appendix B. Particle deposition in SMART model
In this appendix, the expressions used in SMART to estimate the particle deposition are introduced. A detailed explanation and derivation of these expressions are found in [32]. This model does not consider the evolution of pellet ablation, cloud formation or plasmoid drift, as it neglects processes faster than pellet ablation and mass relocation (particle deposition). On the contrary, it considers scaling-laws for the calculation of the pellet ablation, which allow for the calculation of the initial absorbed density, δN, and initial δn; and the size of a cylindrical cloud: • Total pellet particles loss along the pellet trajectory [49]: where γ = 5/3, E ion = 13.6 eV, E diss = 2.2 eV and Λ en = 2T e /7.5. • Toroidal cloud size [50]: where R is the major plasma radius.
Using these expressions, the density deposition shift is calculated directly from the poloidal flux ψ perturbation. Such poloidal flux shift is derived from the current density continuity ∇ · ⃗ j = 0 in equilibrium dV/dt = 0 [51]: Here, B is toroidal magnetic field and q is local safety factor. In addition, β is defined as β = 2µ 0 p/B 2 , where p is plasma pressure. Moreover, a and R are the minor and major plasma radii, <n> = n + <δn> is the flux-surface-averaged density, and <δn ⩾ δn δa L c /4π 2 aR, δN = δn δa 2 L c is the number of the ablated pellet particles. Furthermore, δa and toroidal angle δφ are the pellet sizes in the radial and toroidal directions, respectively, i.e. δa = r 0 and δφ = L c /qR. Finally, ϑ is the angle between the outward normal to the magnetic surface and the direction of the major radius.